RBSE Solutions Class 11 Maths Chapter 9 Logarithms Exercise 9.3

Question 1. Find the characteristic of logarithm of following numbers:
(i) 1270
(ii) 20.125
(iii) 7.985
(iv) 431.5
(v) 0.02
(vi) 0.02539
(vii) 70
(viii) 0.000287
(ix) 0.005
(x) 0.00003208
(xi) 0.000485
(xii) 0.007
(xiii) 0.0005309

Answer:
(i) For the number 1270, the whole number part has 4 digits. To find the characteristic, we subtract 1 from the number of digits. So, the characteristic is \( 4 - 1 = 3 \).
(ii) For the number 20.125, the whole number part (20) has 2 digits. So, the characteristic is \( 2 - 1 = 1 \).
(iii) For the number 7.985, the whole number part (7) has 1 digit. So, the characteristic is \( 1 - 1 = 0 \).
(iv) For the number 431.5, the whole number part (431) has 3 digits. So, the characteristic is \( 3 - 1 = 2 \).
(v) For the number 0.02, there is 1 zero right after the decimal point before the first non-zero digit. When the number is less than 1, we count the number of zeros immediately after the decimal. So, the characteristic is \( -(1 + 1) = -2 \). This is often written as \( \overline{2} \).
(vi) For the number 0.02539, there is 1 zero right after the decimal point before the first non-zero digit. So, the characteristic is \( -(1 + 1) = -2 \) or \( \overline{2} \).
(vii) For the number 70, it is a 2-digit whole number. So, the characteristic is \( 2 - 1 = 1 \).
(viii) For the number 0.000287, there are 3 zeros right after the decimal point before the first non-zero digit. So, the characteristic is \( -(3 + 1) = -4 \) or \( \overline{4} \).
(ix) For the number 0.005, there are 2 zeros right after the decimal point before the first non-zero digit. So, the characteristic is \( -(2 + 1) = -3 \) or \( \overline{3} \).
(x) For the number 0.00003208, there are 4 zeros right after the decimal point before the first non-zero digit. So, the characteristic is \( -(4 + 1) = -5 \) or \( \overline{5} \).
(xi) For the number 0.000485, there are 3 zeros right after the decimal point before the first non-zero digit. So, the characteristic is \( -(3 + 1) = -4 \) or \( \overline{4} \).
(xii) For the number 0.007, there are 2 zeros right after the decimal point before the first non-zero digit. So, the characteristic is \( -(2 + 1) = -3 \) or \( \overline{3} \).
(xiii) For the number 0.0005309, there are 3 zeros right after the decimal point before the first non-zero digit. So, the characteristic is \( -(3 + 1) = -4 \) or \( \overline{4} \).
In simple words: The characteristic tells us about the size of the number. For numbers 1 or greater, count the digits before the decimal and subtract one. For numbers less than 1, count the zeros after the decimal but before the first non-zero digit, add one, and put a minus sign in front.

🎯 Exam Tip: Remember that for numbers greater than or equal to 1, the characteristic is \( (n-1) \), where \( n \) is the number of digits in the integral part. For numbers less than 1, it's \( -(n+1) \), where \( n \) is the number of zeros immediately after the decimal point.

Question 2. Find the logarithm (characteristic and mantissa) of the following numbers:
(i) 2813
(ii) 400
(iii) 27.28
(iv) 9
(v) 0.678
(vi) 0.0035
(vii) 0.08403
(viii) 0.000287
(ix) 1.234
(x) 0.00003258
(xi) 0.000125
(xii) 0.00003208

Answer:
(i) For 2813:
    Characteristic: The number 2813 has 4 digits in its whole number part. So, the characteristic is \( 4 - 1 = 3 \).
    Mantissa: We look in the logarithm table. Find the row for '28', then the column for '1', which gives 4487. Next, find the mean difference for '3' in the same row, which is 5. Adding these, \( 4487 + 5 = 4492 \). So, the mantissa is 0.4492.
    Therefore, \( \log_{10} 2813 = \text{Characteristic} + \text{Mantissa} = 3 + 0.4492 = 3.4492 \).
(ii) For 400:
    Characteristic: The number 400 has 3 digits in its whole number part. So, the characteristic is \( 3 - 1 = 2 \).
    Mantissa: We look in the logarithm table. Find the row for '40', then the column for '0', which gives 6021. So, the mantissa is 0.6021.
    Therefore, \( \log_{10} 400 = \text{Characteristic} + \text{Mantissa} = 2 + 0.6021 = 2.6021 \).
(iii) For 27.28:
    Characteristic: The whole number part (27) has 2 digits. So, the characteristic is \( 2 - 1 = 1 \).
    Mantissa: We look in the logarithm table. Find the row for '27', then the column for '2', which gives 4346. Next, find the mean difference for '8' in the same row, which is 13. Adding these, \( 4346 + 13 = 4359 \). So, the mantissa is 0.4359.
    Therefore, \( \log_{10} 27.28 = \text{Characteristic} + \text{Mantissa} = 1 + 0.4359 = 1.4359 \).
(iv) For 9:
    Characteristic: The number 9 has 1 digit in its whole number part. So, the characteristic is \( 1 - 1 = 0 \).
    Mantissa: We look in the logarithm table. We use '90' to find the mantissa for '9'. Find the row for '90', then the column for '0', which gives 9542. So, the mantissa is 0.9542.
    Therefore, \( \log_{10} 9 = \text{Characteristic} + \text{Mantissa} = 0 + 0.9542 = 0.9542 \).
(v) For 0.678:
    Characteristic: There are no zeros between the decimal point and the first non-zero digit (6). So, the characteristic is \( -(0 + 1) = -1 \) or \( \overline{1} \).
    Mantissa: We look in the logarithm table. Find the row for '67', then the column for '8', which gives 8312. So, the mantissa is 0.8312.
    Therefore, \( \log_{10} 0.678 = \text{Characteristic} + \text{Mantissa} = \overline{1} + 0.8312 = \overline{1}.8312 \).
(vi) For 0.0035:
    Characteristic: There are 2 zeros between the decimal point and the first non-zero digit (3). So, the characteristic is \( -(2 + 1) = -3 \) or \( \overline{3} \).
    Mantissa: We look in the logarithm table. Find the row for '35', then the column for '0', which gives 5441. So, the mantissa is 0.5441.
    Therefore, \( \log_{10} 0.0035 = \text{Characteristic} + \text{Mantissa} = \overline{3} + 0.5441 = \overline{3}.5441 \).
(vii) For 0.08403:
    Characteristic: There is 1 zero between the decimal point and the first non-zero digit (8). So, the characteristic is \( -(1 + 1) = -2 \) or \( \overline{2} \).
    Mantissa: We look in the logarithm table. Find the row for '84', then the column for '0', which gives 9243. Next, find the mean difference for '3' in the same row, which is 2. Adding these, \( 9243 + 2 = 9245 \). So, the mantissa is 0.9245.
    Therefore, \( \log_{10} 0.08403 = \text{Characteristic} + \text{Mantissa} = \overline{2} + 0.9245 = \overline{2}.9245 \).
(viii) For 0.000287:
    Characteristic: There are 3 zeros between the decimal point and the first non-zero digit (2). So, the characteristic is \( -(3 + 1) = -4 \) or \( \overline{4} \).
    Mantissa: We look in the logarithm table. Find the row for '28', then the column for '7', which gives 4579. So, the mantissa is 0.4579.
    Therefore, \( \log_{10} 0.000287 = \text{Characteristic} + \text{Mantissa} = \overline{4} + 0.4579 = \overline{4}.4579 \).
(ix) For 1.234:
    Characteristic: The whole number part (1) has 1 digit. So, the characteristic is \( 1 - 1 = 0 \).
    Mantissa: We look in the logarithm table. Find the row for '12', then the column for '3', which gives 0899. Next, find the mean difference for '4' in the same row, which is 14. Adding these, \( 0899 + 14 = 0913 \). So, the mantissa is 0.0913.
    Therefore, \( \log_{10} 1.234 = \text{Characteristic} + \text{Mantissa} = 0 + 0.0913 = 0.0913 \).
(x) For 0.00003258:
    Characteristic: There are 4 zeros between the decimal point and the first non-zero digit (3). So, the characteristic is \( -(4 + 1) = -5 \) or \( \overline{5} \).
    Mantissa: We look in the logarithm table. Find the row for '32', then the column for '5', which gives 5119. Next, find the mean difference for '8' in the same row, which is 11. Adding these, \( 5119 + 11 = 5130 \). So, the mantissa is 0.5130.
    Therefore, \( \log_{10} 0.00003258 = \text{Characteristic} + \text{Mantissa} = \overline{5} + 0.5130 = \overline{5}.5130 \).
(xi) For 0.000125:
    Characteristic: There are 3 zeros between the decimal point and the first non-zero digit (1). So, the characteristic is \( -(3 + 1) = -4 \) or \( \overline{4} \).
    Mantissa: We look in the logarithm table. Find the row for '12', then the column for '5', which gives 0969. So, the mantissa is 0.0969.
    Therefore, \( \log_{10} 0.000125 = \text{Characteristic} + \text{Mantissa} = \overline{4} + 0.0969 = \overline{4}.0969 \).
(xii) For 0.00003208:
    Characteristic: There are 4 zeros between the decimal point and the first non-zero digit (3). So, the characteristic is \( -(4 + 1) = -5 \) or \( \overline{5} \).
    Mantissa: We look in the logarithm table. Find the row for '32', then the column for '0', which gives 5051. Next, find the mean difference for '8' in the same row, which is 11. Adding these, \( 5051 + 11 = 5062 \). So, the mantissa is 0.5062.
    Therefore, \( \log_{10} 0.00003208 = \text{Characteristic} + \text{Mantissa} = \overline{5} + 0.5062 = \overline{5}.5062 \).
In simple words: To find the logarithm of a number, first figure out the characteristic (the whole number part) based on its size. Then, use a logarithm table to find the mantissa (the decimal part). Combine these two parts to get the full logarithm value.

🎯 Exam Tip: Always make sure to correctly identify the characteristic, especially for numbers less than 1, where it is negative. The mantissa is always positive and found from the log tables, so correctly combine it with the characteristic.